# Modular Function

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A modular function is a meromorphic function on ℍ (the set of quaternions, upper half-plane) which is meromorphic at the cusps, or infinity.

The elliptic lambda is the fundamental modular function, taking values 0, 1, and infinity on the cusps.

The term “modular” comes from the moduli space of complex curves (a.k.a. Riemann surfaces) of genus 1 (Zagier, 1991).

One example of a modular function of weight 0.

An example of a modular function is the j-invariant. In fact, every modular function is a rational function of the j-invariant (Snowden, 2020).

## Formal Definition of a Modular Function

A modular function can be defined in several ways. One way is to view these as complex-valued functions on ℍ which are (Zagler, 1991):

• Invariant under the action τ ↦ (aτ + b)/(cτ + d) of Γ1 on ℍ,
• Holomorphic on ℍ,
• Satisfy suitable growth conditions at infinity.

A modular function can also be defined as one with the following properties:

1. f(μ(z)) = f(z) for all μ in the modular group Γ.
2. f has a Fourier expansion of the form

.

## References

Apostol, T. (1997). Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag.
Borwein, J. & Borwein, P. (1987). “Elliptic Modular Functions.” §4.3 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 112-116.
Milanovich, D. (2020). Modular Functions and Picard’s Little Theorem. Retrieved November 12, 2020 from: https://sites.math.washington.edu/~morrow/336_20/papers19/Daniel.pdf
Prasad, D. (2000). Introduction to modular forms. Retrieved November 12, 2020 from: http://www.math.tifr.res.in/~dprasad/mf2.pdf
Rankin, R. (1997). Modular Forms and Functions. Cambridge University Press.
Schoeneberg, B. (1974). Elliptic Modular Functions: An Introduction. Berlin: New York: Springer-Verlag.
Snowden, A. (2020). Lecture 13: Modular forms. Retrieved November 12, 2020 from: http://www-personal.umich.edu/~asnowden/teaching/2013/679/L13.html
Zagler, D. (1991). Modular Forms of One Variable. Retrieved November 12, 2020 from: https://people.mpim-bonn.mpg.de/zagier/files/tex/UtrechtLectures/UtBook.pdf

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Stephanie Glen. "Modular Function" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/modular-function/
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